L(s) = 1 | − 2i·5-s + (−2.23 − 1.41i)7-s − 1.41·11-s + 4.47·17-s + 6.32·19-s − 3.16i·23-s + 25-s + 3.16·29-s + 5.65i·31-s + (−2.82 + 4.47i)35-s − 4.47i·37-s + 4.47·41-s − 4i·43-s − 8·47-s + (3.00 + 6.32i)49-s + ⋯ |
L(s) = 1 | − 0.894i·5-s + (−0.845 − 0.534i)7-s − 0.426·11-s + 1.08·17-s + 1.45·19-s − 0.659i·23-s + 0.200·25-s + 0.587·29-s + 1.01i·31-s + (−0.478 + 0.755i)35-s − 0.735i·37-s + 0.698·41-s − 0.609i·43-s − 1.16·47-s + (0.428 + 0.903i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.482877030\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.482877030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + 3.16iT - 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 9.48iT - 71T^{2} \) |
| 73 | \( 1 + 6.32iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 6.32iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.221976337363342646417893917815, −7.47744123247292313501537384404, −6.85475800446171513546329668434, −5.88225565550699355598215276537, −5.22180814262161218599741401054, −4.50435573087389426906567654020, −3.48484863478750552753629738644, −2.85836646408021918306616744900, −1.38293303929492346929253564356, −0.49388014411569350700394108157,
1.15775240339099621591368804494, 2.67664573973429509054950427549, 3.02133000744937370686836278481, 3.88724341856587936064171905033, 5.13657924917586522959712174371, 5.76085018741126345766517806701, 6.44200911991656672159387569900, 7.27790224353439345325488611746, 7.75358427469186257540995908807, 8.680758658735089942155308149966